First, with those questions I would highly recommend she get in touch with a divorce lawyer and a loan officer if you have not already. They are familiar with these kinds of situations and can give you real data vs. making decisions off assumptions.
First recommendation, you need to analyze whether you are even allowed to rent out the house to multiple tenants. Many single family homes cannot be rented out by the room due to zoning restriction. Look on the forums here on Bigger Pockets, and start to educate yourself on what it would be like to be a landlord before you intentionally step into that position. If your mom is planning on renting a room out to help pay for her loan, you'll need this information too.
Assuming that this is a 50/50 split of assets, then your dad has a claim to 50% of the equity in the property. He could get his money by having your mom refinance the house at $247k (equity split + remainder owed on the property), but she would need to be able to qualify for that loan. Don't move forward on the assumption that she can't qualify, she (and you) need to talk with a loan officer who can give you real data on which you can base your decisions. If she can't qualify for the loan by herself, then you could be on the loan with her and see if both of you together would qualify. Keep in mind, this now means you are also liable for the payments. So if she stops paying, and you can't afford the extra payments yourself, your credit score will tank and you'll have a foreclosure on your record.
Best of luck to you and her on this journey :)